Equivariant cutting and pasting of \(G\) manifolds (Q1583803)

To study cutting and pasting of compact manifolds with boundary, \textit{H. Koshikawa} [Kyushu J. Math. 49, No. 1, 47-57 (1995; Zbl 0846.57025)] defined the \(SK\)-group \(SK_*(X, A)\) of singular manifolds with boundary in a topological pair \((X, A)\). The paper under review considers the equivariant \(SK\)-group \(SK_*^G(\text{pt}, \text{pt})\), where \(G\) is a finite abelian group. \(SK_*^G(\text{pt}, \text{pt})\) is the \(SK\)-group of compact \(G\)-manifolds with boundary. The set of additive homomorphisms \(SK_*^G(\text{pt}, \text{pt}) \to {\mathbf Z}\) becomes a free \({\mathbf Z}\) module. The author gives a basis for this module. Using this, he gives a necessary and sufficient condition for two \(G\)-manifolds \(M\), \(N\) with boundary to be \(G\)-\(SK\) equivalent in \(SK_*^G(\text{pt}, \text{pt})\). The condition is given in terms of the Euler characteristics of the parts \(M_{\sigma}\), \(N_{\sigma}\) of \(M\), \(N\), where \(M_{\sigma}\) is the part of \(M\) whose points have a slice type containing a given type \(\sigma\). The author also studies a relation between \(SK_*^G\) and \(SK_*^G(\text{pt}, \text{pt})\), and obtains the same kind of condition in \(SK_*^G\) modulo torsion, which was suggested by \textit{C. Kosniowski} [Actions of finite abelian groups (1978; Zbl 0373.57018)]. Finally the author obtains a new generator for \(SK_*^G(\text{pt}, \text{pt})\) modulo torsion, and compares this with the generator already obtained by \textit{T. Hara} and \textit{H. Koshikawa} [Kyushu J. Math. 51, No. 1, 165-178 (1997; Zbl 0884.57028)].